Added Resistance in Waves

TECHN$CHE WIVERSiTT

Laboratorlum Archief Meketweg 2, 2628 CD Deift Tel.:015 786873 -Fax: 015. 781833 PAPER NO. 8

ADDED RESISTANCE IN WAVES Odd M. Faltinsen Professor, Norwegian Institute of Technology, Trondheim

Knut J. Minsaas Head of Research, Norwegian Hydrodynamic Laboratories, Trondheim

ABSTRACT The senstitivity of added resistance to ship form and Froude nuxnber in the ship motion range is discussed. This is done by simplified

analysis, physical reasoning and a review of the published literature. An analytical model capable of predicting the shaft horse power in

irregular waves whilst taking into account the effect o.f the change of propeller characteristics in waves, is presented. This model is used to evaluate the influence of ship length, ship form, propeller sub-mergence and diameter, Froude number and wave direction.

ADDED RESISTMCE IN WAVES

0.M. FALTINSEN and K.J. MINSAAS

I. INTRODUCTION

Added resistance of a ship in regular waves is often studied in two main wave length ranges. This is illustrated in Figure.1. For short waves

(AlL < 0.5) and a head sea, the added resistance is due to the reflection

of waves, in the bow. _{For wave lengths of the order of magnitude of the}

ship length a peak in the added resistance curve occurs as a result of the large relative vertical motion between the ship and the waves.

RAw

pg B2/ L

Due to bow

I

_{Due to}

1 _I.

wave refLectiont

i

shipmotions

I I

0.5

1.0

Head sea

Figure 1 Added resistance due to regular waves.

The added resistance of a ship in waves is closely related to the ship's ability to create waves. _{If the ship does not cause waves as a result of} the ship's motion or the diffraction of the incident waves around the ship, the added resistance will be zero according to potential theory. This is readily appreciated on inspection of Gerritsma and Beukelmants(1) formula, where information on the longitudinal distribution of the sectional wave damping coefficients in heave are required. If the ship does not create waves, the wave damping coefficients are zero and Gerritsma and Beukelman's formula will give zero added resistance. _{This situation occurs in potential} theory when the wave length is large relative to the. ship length. _This does not mean that the real added resistance will be zero in this case. it is evident that a small cork in waves will drift. _{This means that a} ship too will experience an added resistance when the wave lengths are

large. _{Since this cannot be explained by the ship's ability to create}

waves as a result of its motion, we shall explain it by considering the

8-1

second order steady horizontal velocity component of fluid particles of the incident waves. _{This approach will provide an estimate of the added} viscous resistance on the ship due to waves.

Speed predictions, hull form and propeller design are normally based on resistance and propulsion data for still water, even if it is known that waves may represent an essential added resiStance. _{But the added}

resistance in waves is sensitive to the seaway. So in order to judge the importance of added resistance compared with still water resistance, we must also know the percentage of time the ship spends in _{areas of} different sea states. _{In this paper we will try to give a quantitative} measure of the importance of added resistance in _{waves relative to still} water resistance for different ship forms. _{This will be done by comparing}

the shaft horse power in waves with the shaft horse power in still water. Our calculation scheme differs from other _{methods in the way we incorporate}

the effect of the change of propeller characteristics in waves and in the way the effect of an irregular sea is taken into accOunt.

It is not only important to determine the magnitude of added resistance. In the context of fuel economy it is also important _{to know how sensitive} added resistance is to changes in hull form and ship speed and how much power can be saved by reducing the added resistance. _{We will try to} discuss the sensitivity of added resistance.

2. INFLUENCE OF PARATER VARIATION ON ADDED _RESISTANCE

The added resistance in short waves has received special attention lately. The reasons for this are the realisation _{that the addEd resistance, of a} ship can beof importance _{even for smaller waves and that smaller waves} are situations that larger ships frequently meet. _{The "small wave" or} "bow wave reflection" range has been discussed _{by Minsaas, Faltinsen} and Persson(2). _{If a one-parameter Pierson-Moskowitz spectrum is used,} it is found that the "small _{wave length" case corresponds to significant} wave heights Hj,3 < O.0065L, where L denotes the ship length. _{For a} 300m long ship this means that H113 < 1.95m. In the North Atlantic this occurs about 40% of the time. _{For a lOOm long ship small waves correspond} to H1,3 < O.65m, which occurs less than 4% of the time in the North

Atlantic.

In the tsmallwavetI case the added resistance in a head sea is very sensitive to the form of the waterline area in the bow. The added resistance is linearly dependent on the beam B, but _{independent of the}

length and the draft of the ship. _{It is sensitive to theFroude number} and strongly dependent on the significant wave height.

On the other hand Reference (2) concluded that changes in the hull form introduced to minimize added resistance in the "small _{wave length" case} did not provide a signficant saving in fuel consumption.

The added resistance in the "ship motion!' range is an order of magnitude larger than the added resistance in the"bow wave reflection" range. It

is difficult, without extensive _{computer runs or model tests, to specify} quantitatively how added resistance depends on main ship form parameters, Froude number, wave heading and wave length in the "ship motion" range. We will thetef ore, try to do this in a qualitative manner. _{In particular} we will discuss how sensitive added resistance is to changes _{in hull form'}

and Froude number. _{The approach we will follow is a combination of'} simplified analysis and physical. reasoning. _{Finally we will examine our'} conclusions using known numerical and experimental _{data available in thE} open literature.

As a starting point we consider the formula for added resistance in the "ship motion" range proposed by Faltinsen et äi3. This formula was derived by a direct pressure integration approach and is valid for regular sinusoidal incident waves. When the added resistance is at its maximum, a dominant contribution in the formula is the relative motion term

- _dS .. (1)

where the integration is with respect to the water line curve c, n1 is the longitudinal component of the outward normal to c, ç is the relative motion on c, p is the mass density of water and g is the acceleration of gravity.

In the case of a ship with a constant cross-section, we can re-express this formula in the form

B'2

2

1 2

'RB

RA

where B is the beam of the ship and and are the relative motions at the bow and the after perpendicular respectively. This formula can also be applied as a crude approximation for estimating the added resistance of a real ship. Normally it would not be recommended for accurate calculations, but here it will be used as a basis for

discussions of the influence of parameter variations in a qualitative

manner. . . -.

This formula does not imply that the added resistance is linearly dependent in B, since the relative motion is also dependent on B. We will now discuss which parameters influence the vertical relative motion of the heave and pitch of the ship. In order to do this we choose a simplified mathematical model and consider first a ship of constant cross-section in a head. sea at zero Froude number. Repeating our statement that we do not recommend use of this model for accurate calculations, we will use strip theory and a long wave length approximation for the wave excitation

loads. The latter implies that the vertical wave excitation load on a strip is written as the sum of the vertical Froude-Kriloff force and a vertical diffraction force consisting of two parts. The first part is the cross-sectional added mass in heave multiplied by the vertical incident wave acceleration at a representative point of the strip. The second part is the cross-sectional damping in heave multiplied by the vertical incident wave velocity at a representative point of the strip. By following this approach we will find that the amplitu4e of the

vertical wave excitation force in a head sea, l3I Head sea, can be related to the amplitude of the vertical wave excitation force in a beam sea, l3l Beam sea, by the formula

2 i kL

I3I

Head sea = 1F31 _{Beam sea}

E

Again L is the length of the ship and k 2,rr/A, where A is the wave length

of the incident waves. Equation (3) indicates that in a head sea the wave loads along the ship may counteract each other so as to cancel the wave excitation loads on the ship. Equation (3) also implies that the heave amplitude,1n31 Read sea' in a head sea can be related to the heave amplitude

in a beam sea, Ifl31 Beam sea, by the formula

2 kL

In I =

In!

. iii:

sin ()I

(4)

3 Head sea 3 Beam sea

8-3

(2)

Similarly, we may consider the amplitude of the pitch angle

Irij.

If we assume a constant mass distribution along the length of the ship we can write

1151

Head sea = 11131 _{Beam sea}

I{-

cos () + -.--sin()}I

(5)

The analysis of the ship at finite Froude number becomes more complicated. We will in this context only incorporate the important _{effect of the}

frequency of encounter,

e' which is written as

w=w+1

u

₍₆₎

where w02/g = 2/A _{k and U is the forward speed of the ship.}

If we assume the same natural frequency at forward speed as in zero speed, equation (6) indicates that it is an incident wave of larger length that creates

resonance in heave and pitch at forward speed than at zero speed. In

practice for a real ship form, the cancellation _{effect on the heave wave} excitation force may be dominant around the natural frequency for heave and pitch at zero forward speed. _{Due to the effect of the frequency of} encounter, the cancellation effect on the heave and pitch wave excitation loads will be less pronounced at forward speed. The consequences are increased heave and pitch at resonance with increasing Froude number. The latter is true in reality only within certain _{limits of Froude}

number. _{The effect of increasing Froude number may be quite pronounced.} From equations (4) and (5) we see the relevance of discussing the sensitivity of heave in a beam sea on main hull parameters when we discuss the heave and pitch in a head sea. _{We will use heave at resonance in beam sea as} a basis for this discussion. _{If we use Newman's(4) formula for the wave} excitation loads we find that at heave resonance

g p 1/2

Beam sea - a v (2D)

w B33

Here-ca is the undisturbed wave amplitude, w the circular frequency of oscillation and B33(2D) is the cross-sectional two-dimensional damping in

heave. _{We also know that Lewis-form technique provides a satisfactory}

method of finding the added mass and damping coefficients of most of the cross-sectional forms of ships. _{This implies that the beam-draft ratio} B/T and the sectional area coefficient _{= A/BT are sufficient hull} parameters to determine the added mass and damping coefficients; _{a well} known exception being bulb sections... By using results from, say, Tasai(5) and vugts(6) for the added mass and damping coefficient in heave, we can evaluate equation (7). _{We should note that w which is the natural}

frequency and depends on the added mass coefficient, will not be the same for different cross-sectional forms of our ship. The result for heave

at resonance in beam sea as a function of beam-draft ratio B/T and sectional area coefficient _{= A/BT is plotted in Figures 2 and 3. We can conclude}

that the heave motion decreases with increasing beam-draft ratio and increases with increasing sectional area coefficient. In reality the sectional area coefficient is the same as the block-coefficient for our particular choice of ship.

N

at resonance I I I

I

I

I

I

(o.-.'1.0)

0

Lewis form

O\

= 0.94) Ui beam-dratt ratio .4_.4.4

4.---x-- Frank Closefit

B/T

2.0 1.0 113

0Iat

resonance 0.5 1.0

I 0.666

21

Figure 2 _{Heave at resonance in beam sea as a function of Figure 3}

Heave at resonance in beam sea as a function of sectional area coefficient

From the discussion presented we may be tempted to conclude that as long as the cancellation effects of the wave loads along the ship are not

pronounced, the heave and pitch motions in a head sea at resonance decreases by an increasing relative damping with increasing beam-draft ratio and

decreasing block-coefficient. On the other hand, if the cancellation effects are pronounced, we should be careful in making such a statement. The reason is that the natural periods in heave and pitch are dependent on the hull parameters and we will get different cancellation effects at the heave and pitch natural periods of the different hulls. In addition the heave and pitch motions increase with increasing Froude number within certain limits. The conclusions above are based on discussions of

transfer functions.

If we combine the transfer functions with a sea spectrum, it is.obvious that the root mean square values of the vertical motions will depend on the ship length. In addition it is expected that the vertical ship motions as a general trend decrease with increasing ship length. We do not say that this always occurs this depends on the wave spectrum and the ship

length..

The vertical ship motions may depend on main hull parameters other than

CB., B/T and L. Since the waterplane area coefficient Cv can be thought of

representing the local beam-draft ratio along the ship, we may be tempted to conclude that increasing C and keeping the draft constant means

decreasing heave and pitch motions. It would be speculative, based on our simplified theoretical model, to include, any new parameters. But we

should be aware that there is no coupling between heave and pitch in our

simplified model.. One reason is that we have selected a ship model with

fore and aft symmetry. It is known that coupling effects between heave and pitch are of some importance. It would therefore be logical to

introduce .a hull parameter characterizing the fore and aft symmetry of the hull. This may, for instance, be achieved using Loukakis, Perakis and Papoulias'(7) method. They used LCB and LCF, where LCB and LCF are the longitudinal location of the centroids of the waterplane area and the sectional area curve from amidships. Loukakis et al showed that LCB and LCF had an influence on the added resistance.

If we compare our conclusions with what Scbmitke and MurdeyC8) report, based on a systematic investigation of frigate hulls indifferent sea states, we are in agreement concerning the influence of L, B/T,C and CB. In

addition they state that increasing L2/BT at constant displacement has a beneficial effect. But they claim that this is mainly due to the associated increase in ship length. Schmitke and Murdey also report that an increase in beam and draft at constant length has a small detrimental effect. It is important to stress, in the same way as Schmitke and Murdey do, that in any systematic series important parameters should only be included once. Also one should be careful in generalizing the results to ships with. character istics outside the paramter ranges considered.

Gerritsma, Beukelman and

Glansdorp9

carried out an interesting series of tests and calculations in regular waves where the length, draft and block coefficient was kept constant. The beam and the Froude number were varied. TheIr results seem to be in agreement with our conclusions that heave and pitch at resonance decrease with increasing beam-draft ratio and increase with increasing Froude number. From their figures of added, resistance, one.

could be misled 'to think that added resistance is strongly decreasing with increasing beam-draft ratio. But this misinterpretation is due to the way added resistance is normally non-dimensionalized. If we, for instance, use

the length instead of the beam as a non-dimensionalizing length, added resistance seems not to be very sensitive to increasing beam-draft ratio.

This may be explained by our simplified formula for added resistance (see equation (2)) where there is a linearly increasing term with the beam which

counteracts a relative motion term which decreases with increasing beam-draft ratio. _{We may note that our discussion of the. influence of the} beam-draft ratio is based on a comparison of ships with the same draft. If we compare ships with the same beam at different beam-draft ratios, we expect that added resistance decreases with increasing beam-draft ratio.' Loukakis and. Chryssostimidis(10) have presented extensive seakeeping tables based on theoretical calculations for Series 60 ships. They varied CB from 0.55 to 0.9, BIT from 2 to 4, L/B from 5.5 to 8.5 and Froude number up to

0.3. The results are for an irregular sea in which a one parameter Pierson-Moskowitz spectra has been used. From their tables it does not appear that the added resistance is very sensitive to small changes in the ship hull parameters. _{Both the results by Loukakis and ChryssostimidisElO)} and Strom Tejsen et a101) show a strong influence of Froude number on added resistance. This may be. expected by analyzing equation (2) and noting the strong influence of Froude number on the relative motion.

Strom Tejsen et al(11) have pointed out the strong sensitivity of added resistance to the sea spectrum. A two parameter Pierson-Moskowitz spectrum does not seem accurate enough to use if significant wave height and mean period are given. Measured sea spectrum are desirable. On the other hand,

this may not be necessary if we want to compare alternative designs of hull

forms. Since each Beaufort number can corresod to several possible Hj,3 and .T2-values (see for instance Fukuda et al2'), a simple relation

between Beaufort number and added resistance in waves seems impossible. It has also been reported in the literature that the vertical ship motions depend on the bulb, transom stern hull, and the pitch radius of gyration. Furthermore, ships with U-form sections seem to have different heave and pitch motions compared with ships with V-form. Our simplified theoretical model is not able to explain this. Generally speaking it does nOt appear that the added resistance in the "ship motion" range is sensitive to small changes in the main hull parameters. However,, there are exceptions which we will present later.

If we want to reduce the added resistance by changing the waterplane-area coefficient, the beam-draft ratio, the length-beam ratio or the block coefficient, we may very well be in a situation where the resulting anticipated changes in added resistance are within the uncertainty in predicting added resistance. In a discussion to the paper by Faltinsen et al'3), Gerritsma pointed out that substantial differences may occur in th'eoretical predictions of added resistance in the "ship motion" range depending on what combinations of ship motion theory and added resistance theory are used. _{We should also be aware that hull form changes that} decrease added resistance may increase still water resistance.

Since the heave and pitch motions may be larger at a high ship speed than at a low ship speed, the sensitivity of added resistance to main hull parameters may be more 'pronounced for high speed ships. _{Furthermore, the}

sensitivity of added resistance may be larger than the sensitivity of the heave and pitch motions. This is due to the quadratic dependence of added resistance on the relative vertical motion (see equation (1)) and that the relative motion may be more sensitive than the amplitude of heave and' pitch motion. We say "may be" about the larger sensitivity of added

resistance because it is based on not varying the beam. In addition, heave and pitch may contribute differently to the relative motion and have

different sensitivities to hull parameters.

3. EFFECT OF THE CHANGE OF PROPULSIVE FACTORS IN WAVES

When we want to evaluate the influence of added resistance on engine power, we should also take into account that the propulsive factors chane in waves. This is not normally done. As shown by Moor and Murdey(hi) and

later by Nakamura and Naito(14), the ship motions will decrease the wake and the thrust deduction. The reductions in wake and thrust deduction are at a maximum when pitch motions are at a maximum. This is also

discussed in Faltinsen et al(3) and will result in decreased hull efficiency and increased propeller efficiency. There are also indications that the relative rotative efficiency changes in a seaway. However, we feel that

the total picture is too complicated for a theoretical treatment. Further-more, these changes are small for moderate sea states. For simplicity we have therefore only studied changes in the propeller efficiency in

subsequent sections and this is probably the weakest part of our study. In Reference (3), Faltinsen et al, and in Reference (2), Minsaas et al discussed a method for estimating the thrust diminution due to restricted propeller shaft immersion in open water. The diminution due to the wave

set up by the propeller and the diminution due to the emergence of the propeller disk were taken into consideration. A thrust diminution factor

8 was developed. This factor is independent of the advance ratio Jo. If

KTO and KQo are the thrust and torque coefficients for a deeply submerged propeller, the values at restricted immersion at the same J0 value will be

= B

ro and KQ

8m KQ0

where m is a constant between 0.8 and 0.85.

A

procedure described in Reference 2 to obtain B by Kempf(14) shows good agreement.

The B curve can be approximated as

I - 0.675(1 - O.769(h/R))1258 , h/R < 1.3

(8)

comparison between the and experimental results

, h/R > 1.3

Here h is the instantaneous propeller shaft immersion and R is the propeller radius.

We shall apply this simple relation for corrections of the propeller characteristics in waves. We will first study a propeller operating behind a ship in regular waves. The instantaneous values of h/R can then be written as

h _{- o} a e

RR

R

h s sin Cu

t+)

(JO)

where Sa is the wave amplitude and c is a phase angle of the relative vertical motion between the ship and the free surface at the propeller. It is difficult to establish coupled relations between instantaneous values of revolutions, immersion, thrust, torque and speed for a ship moving

through the waves. For simplicity we will average torque, thrust and resistance in time. In doing this we will assume that the influence of variation of J0 during one wave cycle in a regular wave can be neglected. The effect of ventilation is not accounted for. This problem occurs mainly for small J0 values and has been discussed by Minsaas et al in Reference (2).

4. POWER INCREASE IN WAVES

We will now show how we can find the shaft horse power in regular waves by using the approach described above. We will compare our values with shaft horse power in still water under the assumption that the mean speed is the

same in waves as in still water. The results in regular waves will be used to obtain mean values of shaft horsepower in irregular waves.

The thrust and torque coefficient in the propeller diagram can, according to our procedure for regular waves, be written as

and

KQ=mKQO

(12)

The bar indicates mean values over one wave period. We will assume that KTO and Kj can be approximated as

KTO=a+bJ0+cJo2

(13)

and

KQ0=d+eJ0+fJ02

(14)

The average propeller thrust in regular waves can also be written as

R +R

o AW

_kJ2

KT - _{pnZ.D4(l_t) - 0}

where R0 is still water resistance, RAW the added resistance in waves, n the number of propeller revolutions, t the thrust deduction coefficient and D the propeller diameter. Further k is determined by (15) and J0 is given by

_U(1)

nD

where U is the ship speed and w is the wake factor. Intersection between KT andKT gives a second degree polynomial for J0, which can be easily

solved. From equation (12) we can now find K0. If we use the index S for the still water values of KQ, J0 and the haft horse power SHP, we can now write the relative increase in the shaft horse power as

Sm' KQ

(OS)3

Sm'S - KQ _J

This is a result for a regular wave. In order to use these results in connection with an irregular sea prediction, we will first assume that the waves are consistent with a narrow banded process and that they are longcrested. It is then legitimate to cut the wave record into successive regular wave parts with amplitude ttatt and circular frequency w. The

probability density function for a and w for a stationary rocess

(i.e. short term prediction) can, according to Sveshnikov('6), be written

2 a w

-2w1w+w2)

₁

a 2 [ 2( 2 f(a,w) = exp 2 2 2 I (18) 2 2 2a

_(w2w1)

_J

where a varies from 0 to while w varies from -

to +. Physically a

negative w should be interpreted in the same way as a positive w.

as

(15)

8-9

(II)

Furthermore, w1 and w2 are the circular frequency defined by the first and second moment of the wave spectrum. Instead of and w2

it

is more usual to use the mean period T1 and T2 defined by

T1 = 2r/w1 (19)

and

T2 = 2ir/w2

(20)

For a modified Pierson-Moskowitz spectrum we can write

T1 = 1.086 T2

(21)

Finally 2 _{in (18) is the area under the wave spectrum. In our case this can} be related to2the significant wave height H113 by

2

H113

(22) 16

The procedure above enables us to determine the average SliP in a given sea state, i.e. for a given H113, T2 and wave direction. This can be written as

SHP K0

J03

(s ) = / da f dw f(a,ü) K -J / (23)

save

0 - QS 0

In order to find the average value over several values of H113, T2 and wave direction, we need frequency tables of H1,3, T2 and wave directions. An example of frequency tables for H1,3 and T2 for the North Atlantic is given in Table 1. The long term average values of SHP can be written

as SlIP i k KQ

K0g31

siip5 ave E E I p(H113, T, c )[/ da Ldw f(a,ci)

ç

]

(24)

where p(H1,31, T23, )

is

the probability that a significant iave interval with mean value H1,31, mean period interval with mean value T23 and wave direction interval with mean value occurs.

5. NUMERICAL RESULTS

In this section we have evaluated the increase in shaft horse power in waves. We have varied the ship length, the ship form, the propeller

immersion and diameter, the Froude number and the wave direction. The results are for the North Atlantic and are either presented as a function of significant wave height or as an average over all sea states in the North Atlantic.

We have investigated four ships. Three of these ships correspond to a Series 60 form with Cb = 0.6, 0.7 and 0.8 and the fourth ship is the container hull form S-175, which was used for comparative hull motion calculations by the member organizations of the ITTC. The added resistance in regular waves of these ships at different Froude number, wave heading and wave frequency was calculated by the direct

integration

method of Faltinsen et al3). Some of these results have already been presented in Reference (3). The bow wave reflection effect (see Figure 1) has not been included. The importance of the small wave length effect on the added resistance has been briefly discussed earlier, but is more extensively evaluated by Ninsaas et al. In the context of energy

Wave Frequency in the North Atlantic Ocean for Whole Year according to Walden's Data

Table 1.

Theshi, motions have been calculated by the Salvesen-Tuck-Faltinsen

method"7).

From the heave and pitch motion it, is straightforward _to estimate the relative vertical motion at the propeller. _{The latter} information is needed in our calculation procedure for the wave effect on the propeller.

The following open water diagram was used:

KTO = 0.3834 - 0.3543 Jo - 0.0924 Jo2

and

'ko

= 0.04679 - 0.03013 Jo - 0.02154 Jo2

8-Il

Wave Period (sec) _Sum

over all Periods 5 7 9 11 13 15 17 20.91 11.79 4.57 2.24 0.47 0.06 0.00 0.60 40.64 0.75-72.78 131,08 63.08 17.26 2.39 0.33 0.11 0.77 287.80 1.75-2.75- 21.24 12641 118.31 30.24 3.68 0.47 0.09 0.56 301.00 3.28 49.60 92.69 32.99 5.46 0.68 0.12 0.27 185.09 0.53 16.19 44.36 22.28 4.79 1.14 0.08 0.29 89.66 0.12 4.34 17.30 12.89 3.13 0.56 Ô.13 0.04 38.51 0.07 2.90 9.90 8.86 3.03 0.59 0.08 0.03 25.46 6.75-0.03 1.39 4.47 5.22 1.93 0.38 0.04 0.04 13.50 0.00 1.09 2.55 3.92 1.98 0.50 0.03 0.02 10.09

8.75-975_

0.00 0.54 1.36 2.26 1.54 0.68 0.20 0.04 6.62 0.01 0.01 0.10 0.11 0.10 0.05 0.02 0.00 0.40 > 10.75-0.00 11 75- 0.00 0.03 0.08 0.17 0.06 0.00 0.34 12.75- 0.05 0.00 0.14 0.22 0.06 0.01 0.48 0.02 0.07 0.09 0.03 0.01 0.22 13.75-0.02 0.06 0.02 0.00 0.01 0.11 14.75-0.00 0.02 0.00 0.02 0.02 0.02 0.01 0.01 0.08 15.75- . . Sum over all 118.9.7 345.43 358.72 138.59 29.05 5.63 0.92 2.69 1000.00 Heights

The still water resistance _{was calculated according to a simplified} procedure and the resistance due to wind and steering was neglected. At R0, the propeller is assumed to operate at = 0.4 and = J.0. The ratio between the propeller radius R and the ship length L was normally set

equal to 0.02.

Weather data from the North Atlantic were used (see Table 1). The wave period in the table was interpreted _{as T2 and the wave height as Hj13.} This choice is not obvious. _{In some cases with high sea states in a} following sea, the total resistance becomes negative. _{These cases were} excluded. _{The major part of the results for shaft horse power is presented} as a function of H113. _{In doing this we have averaged the results over all} wave period corresponding to each significant wave height. _{This means that} equation (24) has been used except for the sunation _{over H1,3 and} .

SHP SHPS

2.0

1.5

Figure 4 _{Effect of ship length on the increase of shaft horse} power in waves. NORTH ATlANTIC Fn 0.2 Head sea Series 60, Cb: 0.6 h0/R 1.3

L:lOOm

L:150m

1= 200 m

I

250 m

I: 300 m

H1(m)

p:O.09

Table 2. Average over all sea states in North Atlantic of the relative increase

in shaft horse power' S'HP/SHPs.in waves.

Ship

eng

Case I Cb=O.G a 00 Case II

CbO.G

h0/R1.0h0/R-Case III Cb=O.6 Case IV Cb=O.G

Fn0.3

Case V Cb=O.S a = 00 Case VI Cb=O.7 a = 0° Case VIlCase ITrC-ship Viii Cb=O.6 a = 300 Case IX

CbO.6

a = 600 Case, X

CbO.6

a = 9O Case XI

Cb0.6'Cb=0.6

120 Case XIICase a = 15O°a )fl

CbO.6

= 180° Case XIV Cb=O.6 average

200 m

300 m

1.47 1.11 1.55 1.17 1.41 1.10 1.24 . 1.26 1.06 1.29 1.07

1.33

1.08 1.48 1.12 1.35 1.11 1.02 1.01 1.10 1.04 1.06 1.02 1.04 1.01 1.21 1.06

Case VIII

Case I

Series 60, Cb

Case II

= 0.6, Fn = 0,2, h0/.R = 1.3,

head'

sea

Series 60, Cb = 0.6, Fn = 0.2,

Case IX

Series 60, Cb = 0.6, Fn = 0.2

= = 30°

60°

h

= 1.3, a

= 1.3, a

Series 60, Cb

Case III

= 0.6, Fn = 0.2, h0/R = 1.0, head sea

_{Case X'}

Series 60, Cb = 0.6, Fn

0.2, h = 90 -

= 1.3, a

Series 60,

_Cb=

0.6, Fn

CaseIV

Series 60, Cb = 0.6., Fn

Case V

Series

60, Cb =

0.8, Fn

= = = 0.2, 0.3, 0.2, -'

_{Case XI}

.

Series 60, Cb = 0.6, Fn = 0.2,

Case XII

Series 60,

_{Cb =}

0.6, Fn = 0.2,

Case XIII

h

. = 1.3, a = 1.3,' a = = 120° 150° ,

head sea

h

2

=

1.0, head sea

h0

-= 1.3, head sea

Series

60, Cb = 0.6,

Fn = 0.2,

h

= 1.3, a = 180

Case VI

h0

_{Case XIV}

Series

60,

Cb = 0.7, Fn = 0.2,

-. = 1.3, head sea

Series 60, Cb -

0.6,

Fn - 0.2,

= 1.3,

Case VII

Average, over all wave headings.

All wave heading

In Figure 4 is shown the effect of ship length on the increase in shaft horse power in waves. _{Also indicated in the figure is the probability} for different H113values in the North Atlantic. _{It is evident that the} ratio SHP/SHPS for the smaller ship lengths rises quickly for moderate sea states and soon becomes unrealistically high. _{The reason for the} latter is that the calculations _{are based on the same ship speed in waves} as in still water. _{In reality voluntary speed reduction due to slamming,} water on deck, large accelerations or propeller racing will _{occur for} the larger sea states for the smaller ships. _{For moderate sea states} involuntary speed reduction due to insufficient power may occur. Another way of carrying out our calculations would, of course, be to keep the same power in waves as in still water and study the resulting speed reduction. _{But we did not choose to do this. In any case it has} secondary importance in the study of added resistance in _{waves when} voluntary speed reduction occurs.

In Table 2 an average of SHP/SHPs for all sea states in the North Atlantic is presented. _{This is only carried out for the ship lengths of 200m and}

3COm. _{We have omitted the results for smaller ship lengths because the}

SHP/SHPs ratio becomes unrealistically high.

We should note that the results in Figure _{4 is only for a head sea. This} occurs of course only a fraction of the total time. _{It is therefore also} of interest to investigate the influence of wave direction on the increase of shaft horse power in waves. _{This is presented in Figures 5 and 6 for} the ship lengths of L = 200m and L = 30Cm respectively. _{It does not appear}

that SHP/S}lPs is very sensitive to wave heading variations between cz = 00

and 600. _{The same is true for wave headings between 120° and 1800.} But the results for a following sea are obviously significantly lower than for a head sea. _{It is interesting to note what is happening when averaging} over all wave headings. _{This is illustrated in Table 2 under the assumption} that all wave headings have an equal probability of occurrence. For a

Series 60. ship with Cb = 0.6 at _{'n = 0.2 we found that an average of} SHP/SHP5 for all sea states _{was 1.21 for a 200m long ship and 1.06 for a} 30Cm long ship. _{These numbers indicate that the still water resistance is} of greater importance than the added resistance in waves.

But even if the average effect of the added resistance in _{waves is relatively} small, it may have some interest to quantify the sensitivity of added

resistance to ship form. This is done in Figure 7 for a head sea. Average values of the influence of ship form for a head sea over all sea states in the North Atlantic are presented in Table 2. _{One should bear in mind} that one can be misled by Figure 7 and Table 2 when discussing how the ship form affects the added resistance in waves. The reason is that SHP/SHPS is a ratio between the shaft horse power in waves and the shaft horse power in still water for one particular ship. _{It may have been} better to have expressed SlIP as SlIPS + tSHP and _{measured LSHP for each} ship relative to the shaft horse power in still water for one reference

ship. This is shown in Table 3 for _{= 4.25 m, L = 200m, Fn = 0.2 and} a head sea condition. _{The Series 60 ship with Cb = 0.6 is used as the} reference ship. _{Some of the main particulars of the investigated ships} are also presented in the table.

SHP SHP5 2.0 1.5

8-15

North Atlantic

Fn: 0.2

Series 60, Cb :0.6 h0/R 1.3 L 200 m North Atlantic Fn 0.2 Series 60, Cb 0.6 h0/R 1.3 1 300 m a.: 30°

a.: 00

: 60° : 120° 150° a. :1800 U : 90° H113 Cm)

Figure 5 Effect of wave direction on the increase of shaft horse power in waves

Figure 6 Effect of wave direction on the increase of shaft horse power in waves.

Still water resistance (Still water resistseries6

CbO.6

Series 60 Seriesr6O ITTC ship

Cb=O.8 Cb:O.7 1.52 1.20 1.03 North Atlantic Fn = 0.2 Head sea h0lR = 1.3

--x-- Series 60, Cb = 0.6

-OP--- Series 60, Cb = 0.7

-6--- Series 60, Cb = 0.8

-D---ITTC ship

Table 3. _{Main particulars and example on influence of ship form}

on shaft horse power in.waves.

Ship _Cb

_BIT

_{L(m) B(in)} SHP(H113=4.25 m)

SHP5(Series 60, cb_0.6) S-175 (ITTC-ship) 0.57 2.672 200 29 0.53 Series 60, Cb=O.6 0.6 2.5 200 26.7 0.7 Series 60, Cb0.7 0.7 2.5 200 28.6 0.55 Series 60, CbO.8 0.8 2.5 200 30.8 0.59 0 _{3 4}

_Hi3(m)

- Figure 7 Effect of ship form on the increase of shaft horse power in waves.

We do not find the differences between the Series 60 ships interesting in the context of fuel economy. The differences in ship forms are too

large. However, the difference between the ITTC ship and the Series 60, Cb = 0.6 is interesting. The main particulars are very much the same and

it is difficult, based on our earlier discussion of the sensitivity of added resistance to hull form, to explain the differences. The ITTC ship has a higher beam-draft ratio and a lower block-coefficient than the Series 60 ship. This should indicate lower added resistance for the ITTC ship. On the other hand the ITTC ship is wider and has a smaller wáterplane area coefficient C* than the Series 60 ship. This should indicate a larger added resistance for the ITTC ship. One reason for

the difference in added resistance may be that the ITTC ship has a bulb, while the Series 60 has not. Furthermore the Series 60 ship has a pitch radius gyration of O.25L compared to O.24L for the ITTC ship. But it would be speculative for us to explain the differences based onour

simplified theoretical model. A more accurate analytical model is necessary. We have also carried out, as accurately as possible, a calculation for

the Series 60 ship with a pitch radius of O.24L instead of O.25L. It certainly has an influence. If we average over all sea states,, but consider only a head sea, we found that SHP/SHPs for L = 200m is 1.42 when kyy = O.24L. Similarly for = O.25L we found SHP/SHP5 = 1.47. For the ITTC ship SHP/SHPs is 1.33.

Let us try to evaluate the differences between the ITTC ship and the original Series 60, Cb = 0.6 in the context of fuel economy. We must therefore average over all wave directions. This was done for the Series 60 ship with Cb = 0.6 and the results are presented in Table 2. We have not done this for the ITTC ship. But based on the data for a head sea, it is possible that an average value of S}/SI5 is 1.16 for a 200m long ITTC ship, while it is 1.05 for a.300m long ITTC ship. The difference between the 200m long ships are of interest in the context of fuel

economy. But we should have in mind the uncertainties in the predictions of the added resistance in waves. This has been discussed earlier. When we compare the ITTC ship and the Series 60, Ch = 0.6 ship it is

inter-esting to point out that the differences in aaded resistance is significantly larger than the differences in still water resistance.

We have also investigated the effect of propeller submergence on the increase of shaft horse power in waves. This is illustrated in Figure 8 'for h0/R = 1.0 and 1.3. When h0/R = 1.0 the tip of the propeller at its

-highest point touches the free surface in still water. This may be possible propeller configuration for a ballast condition. When h0/R = 1.3, the

-influence of the propeller is close to the influence of the deeply submerged propeller Figure 8 therefore quantifies the influence of the change of propulsive factors in waves, on the shaft horse power in waves. We should note that the differences between h0/R = 1.0 and 1.3 persists when H113 - 0.

:The reason is the still water wave making effect of the propeller. The average over all sea states in the North Atlantic is presented in Table 2. -These results should indicate that it is beneficial to look for a propeller

system which-will always be deeply submerged.

= 0.70 for the ITTC ship and = 0.73 for the Series 60, Cb '= 0.6.

8-1 7

2.0 1.5 1.5 SHP SHP5

/

/

I,

I,,

/

/

L:100 m North Atlantic Fn : 0.2 Head sea Series 60, Cb 0.6

:0.02

o---h0IR :1.0

--r-- h0/R =1.3

/

I I

/

IC

/

/

/

/

/

/

/

_,

,

/

/

./

/

-

.

-L:100 m I,

I,

/

I

North Atlantic

Fn :0.2

Head sea ITTC ship h,/L :0.026

K

_:0.015 0.020 = 0.025 L:200m

4.

.7

/

L:300 m

-=----1__=

-1.0 0 1

2.

3 4 H113

Figure 9 Effect of propeller diameter _{on the increase of shaft} horse power

in

waves

J.

0 1 2 3 4

Hj3(m)

Figure 8 _{Effect of propeller submergence on the increase of shaft} horse power in waves.

SHP SHPS

2.0

IC

In Figure 9 we have shown the influence of propeller diameter for _constant propeller shaft immersion h0/L = 0.026. It is evident that a large

propeller diameter is more sensitive to wave motions than a small-diameter. In Figure 10 the effect of Froude number on the increase of shaft horse power in waves is shown. _{There is evidently a sensitivity to Froude} number. _{The results indicate a possible saving in fuel consumption by}

slowing down the ship. This is not unexpected. We should note that SHP/SHP5 is a ratio for shaft horse power in waves and still water at the same Froude number. _{We have therefore also indicated in the figure the} ratio between the still water resistance at Fn = 0.3 and Fn 0.2, which are the two Froude numbers we investigated. _{The reason why SHP/SHP is} smaller for Fn = 0.3 than for Fn = 0.2 is that SHPs at Fn = 0.3 is 3.2

times larger than SHPS at Fn = 0.2.

(Stilt water resistance) FnrO.3

_:2.15

(Stilt water resistance) Fn 0.2

8-19 North Atlantic Head sea h0/R :1.0 Series 60, Cb : 0.6

O----Fn :0.2

--x-- Fn :0.3

Figure 10 _{Effect of Froude number on the increase of shaft horse} power in waves

6. CONCLUSIONS

An analytical model that predicts the shaft horse _{power in irregular waves} and tries to take into account the effect of the change of propeller

characteristics in waves as well as the added resistance in waves, is presented. _{The influence of hull form, propeller submergence and Froude} number and wave direction is of importance, in particular for the smaller

Acknowl edexnent

The authors appreciate the discussions with J. Journee and T. Vinje as

REFERENCES

1. GERRITSMA, J. and BEUKELMAN, W

in Waves of a Fast Cargo Ship" Committee, Proc. 13th ITTC Vol

(Sept. 1972) p. 285.

MINSAAS, K. ,FALTINSEN, 0. and PERSSON, B. "On the Importance of Added Resistance, Propeller Immersion and Propeller Ventilation for

Large Ships in a Seaway". PRADS, Tokyo, Oct. 1983.

FALTINSEN, O.,MINSAAS, K.,LIAPIS, N. and SKJORDAL, S.O. "Prediction of Resistance and Propulsion of a Ship in a Seaway". Thirteenth symposium on Naval Hydrodynamics, Tokyo, 1980.

NEWMAN, J.N. "The Exciting Forces on Fixed Bodies in Waves". J. of Ship Res. 6, no. 3, Dec. 1962.

TASAI, F. "On the Damping Force and Added Mass of Ships Heaving and Pitching." Report of Research Institute for Applied Mechanics, Hyushu University, VII, no. 26, 1959.

VUGTS, J.M. "The Hydrodynamic Coefficients for Swaying Heaving and Rolling Cylinders in a Free Surface't. Netherlands Ship Research Centre T.N.O., Report 112s, May 1968.

LOUKAKIS, T., PERAKIS, A.N. and PAPOULIAS, F.A. "The Effect of Some Hull Form Parameters on the Seakeeping Behaviour of Surface Ships." Conference onSeagoing Qualities of ships and Marine Structures,

September 1983, Varna.

SCHMITKE,. R.T. and MUP.DEY,D.C. "Seakeeping and Resistance Trade-Off s in Frigate Hull Form Design." Thirteenth Symposium on Naval

Hydrodynamics, Tokyo, 1980.

GERRITSMA, J., BEUKELMAN, W. and GLANSDORP, C.C. "The Effects of Beam on the Hydrodynamic Characteristics of Ship Hulls". Tenth Symposium on Naval Hydrodynamics, Boston, 1980.

LOUKAKIS, T.A. and CHRYSSOSTIMIDIS, C. "Seakeeping Standard Series for Cruiser-Stern Ships." SNA Trans, vol. 83, 1975.

STROM-TEJSEN, J., YEN, H.Y.H. and MORAN, D.D. "Added Resistance in Waves." SNANE Transactions 1973.

FIJKUDA, J. et al "Estimating the Design Values of Hydrodynamic Pressure

Induced on the Ship Hull in Waves." Trans. of the West-Japan Society of Naval Architects, No. 49, Feb. 1975.

MOOR, D.I. and MURDEY, D.E. "Motions and Propulsion of Single Screw Models in Head Seas.. Part II." TRINA, Vol. 112, 1970.

NAKANURA, S. and NAITO, S. "Propulsive Performance of a Container Ship in Waves". The Society of Naval Architects of Japan, Naval Architecture and Ocean Engineering, Vol. 15, 1977.

"Analysis of the Resistance Increase Appendix 5 of Report Seakeeping

2 (1972) p. 902; ISP,, Vol. 19, No. 217

KEMPF, C.

_{"Inmiersion of Propellers".}

_{Trans. North East Coast}

Institution, 1930.

SVES}INIKOV, A.A.

_{"Applied Methods in the Theory of Random Functions".}

Pergamon Press Ltd. 1966.

SALVESEN, N., TUCK, E.0. and FALTINSEN,

_0.

_{"Ship Motions and Sea}

Nomenclature

a or Sa Wave amplitude

A Cross sectional area

B Ship breadth or beam

Cb or CB Block coefficient

C_w Water plane area coefficient

D _{Propeller diameter}

f(a,w) _{Sveshnikov joint probability distribution of wave}

amplitude and wave frequency (circular)

F1 _{Added resistance force component due to relative motion}

F3

_{Vertical wave excitation force} g _{Gravitational acceleration}

H113 Significant wave height

J Advance ratio

0

k _{Wave number, k = 2ii/X}

Propeller thrust coefficient

KT* Average propeller thrust

KQ Propeller torque coefficient

L Ship length

n Propeller revolutions

n1 Longitudinal outward normal component

p( ) Probability density function

R Propeller radius

R Still water resistance

RAW Added resistance in waves

S _{Subscript indicating still water values of associated variable}

SHP Shaft horse power

t Time or thrust deduction

T Ship draft T1 or T2 Mean periods U Ship speed w Wake factor a Wave direction Phase angle

Amplitude of incident wave Relative ship motion

Relative ship motion at aft perpendicular and bow respectively Heave and pitch amplitude of ship

Wave length p Water density RB T)3, 115 A 8-23

a Sectional area coefficient

a2 Area under wave spectrum (variance of original data) '

Incident wave frequency